class: top, inverse, title-slide .title[ #     Neural Dynamic N-mixture Model ] .subtitle[ ## A deep learning framework for inferring demographic rates from count data ] .author[ ### <strong>Speaker:</strong><br>François Leroy (he/him)<br><strong>Supervised by:</strong><br>Marta Jarzyna<br><br><br><br> ] .institute[ ### <br><br><br><br>The Ohio State University ] .date[ ### 2025-10-07 ] --- # Beyond abundance change <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Demographic rates </h2> </div> .pull-left[ * Changes in abundance are valuable, but demographic mechanisms offer deeper ecological insights * Species' extinctions event are the result of increasing difference between recruitment and loss * Changes in demographic rates can be early warnings of species extinction * Monitoring survival and recruitment can help tailoring efficient conservation policies ] .pull-right[ <br><br> <img src="data:image/png;base64,#images/slide1.png" width="2072" /> ] --- # Beyond abundance change <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Problem </h2> </div> .pull-left[ * Data for demographic rates models are per individual (i.e. *individual encounter history data*) * They are costly in resources and time * Individual identification is not equal for all taxa * Can be invasive/traumatic * Limited spatial and temporal extent ] .pull-right[ <div style="margin-right:-3.5em"> <img src="data:image/png;base64,#images/slide2.png" width="100%" /> ] --- # Beyond abundance change <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Using abundance to infer demographic rates </h2> </div> .pull-left[ * However, abundance data are available at large spatial and temporal scale * The idea of inferring birth/immigration (`\(\mu\)`) and death/emigration (`\(\lambda\)`) from experimental data is not new * While overall change in abundance gives information about `\(\lambda-\mu\)`, the volatility of the time-series provide information about `\(\lambda+\mu\)` making `\(\lambda\)` and `\(\mu\)` identifiable ] .pull-right[ <br> <img src="data:image/png;base64,#images/slide3.png" class="bigimg" /> ] .footnote[Wilkinson, 2011] --- # Dynamic N-mixture model <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Hidden Markov Model </h2> </div> **Observation process** .pull-left[ `$$y_{i,j,t} \sim Binomial(N_{i,t}, p)$$` ] .pull-right[ `\(y =\)` observed abundance<br> `\(N_{it} =\)` latent "true" abundance<br> `\(p =\)` detection probability ] .footnote[Dail & Madsen, 2011] -- **State process** .pull-left[ `$$N_{i,1} \sim Poisson(\lambda)$$` <br> `$$S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\ R_{i,t+1} \sim Poisson(\gamma_{i,t})\\ N_{i,t+1} = S_{i,t+1} + R_{i,t+1}$$` ] .pull-right[ `\(\lambda =\)` abundance at `\(t = 1\)` <br> `\(\phi =\)` survival probability<br> `\(\gamma =\)` number of recruits ] .footnote[Dail & Madsen, 2011] --- # Limitations of Existing Frameworks .pull-left[ **Limitations of hierarchical framework** * Linear or simple polynomial effects of covariates, even though true ecological responses are often unknown *a priori* * MCMC algorithm scales poorly because inherently sequential, little possibility of parallelization * Sensitive to priors and initial values ] .pull-right[ **Limitations of Neural Networks** * Usually lack inferential power, making it less relevant for ecological insights * Doesn't distinguish between ecological process and imperfect detection ] -- .center[ <img src="data:image/png;base64,#images/joseph.png" width="70%" /> ] --- # Neural hierarchical model .center[ <img src="data:image/png;base64,#images/joseph.png" width="90%" /> ] <br> * Combines flexibility and scalability of neural networks with the inferential power of hierarchical models * Combines decades of development in hierarchical modelling for ecological data with the flexibility and predictive power of Neural Networks * Loss function: tailored from the model's specific Likelihood (i.e. negative log-likelihood) * Output activation function according to the parameter to infer --- # Neural Dynamic N-mixture model <!-- .footnote[\* Krishnan et al. 2017] --> <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Hidden Markov Model </h2> </div> .pull-left-narrow[ <br> **State process** `$$\begin{align*} & N_{i,1} \sim Poisson(\lambda) \end{align*}$$` <br> `$$\begin{align*} & R_{i,t+1} \sim Poisson(\gamma_{i,t})\\ & S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\ & N_{i,t+1} = S_{i,t+1} + R_{i,t+1} \end{align*}$$` **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` ] .pull-right-wide[ <div style="margin-right:-3.5em"> <br> <img src="data:image/png;base64,#images/NN_DNM.png" width="2807" /> ] --- # Neural Dynamic N-mixture model <div class="center" style="margin:0;"> <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;"> Hidden Markov Model </h2> </div> .pull-left-narrow[ <br> **State process** `$$\begin{align*} & N_{i,1} \sim Poisson(\lambda) \end{align*}$$` <br> `$$\begin{align*} & R_{i,t+1} \sim Poisson(\gamma_{i,t})\\ & S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\ & N_{i,t+1} = S_{i,t+1} + R_{i,t+1} \end{align*}$$` **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` ] .pull-right-wide[ <div style="margin-right:-3.5em"> <br> <img src="data:image/png;base64,#images/NN_DNM1.png" width="2795" /> ] --- # Example with the N-mixture model **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` **State process** `$$\begin{align*} & N_{i} \sim Poisson(\lambda) \end{align*}$$` -- <br> **Integrated likelihood:** <br> `$$\begin{align*} &\hfill L\!\left(p,\lambda \mid y_{it} \right) = \prod_{i=1}^{R} \left( \sum_{N_i = 0}^{\infty} \left( \prod_{t=1}^{T} Bin \!\left(y_{it}; N_i, p\right) \right) Pois \!\left(N_i;\lambda\right) \right) \end{align*}$$` --- # Example with the N-mixture model **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` **State process** `$$\begin{align*} & N_{i} \sim Poisson(\lambda) \end{align*}$$` <br> **Integrated likelihood:** <br> `$$\begin{align*} &\hfill L\!\left(p,\lambda \mid y_{it} \right) = \prod_{i=1}^{R} \left( \sum_{N_i = 0}^{K} \left( \prod_{t=1}^{T} Bin \!\left(y_{it}; N_i, p\right) \right) Pois \!\left(N_i;\lambda\right) \right) \end{align*}$$` <br> * Marginalize over `\(K >> y_{it}\)` --- # Dynamic N-mixture model **State process** `$$\begin{align*} & N_{i,1} \sim Poisson(\lambda)\\ & R_{i,t+1} \sim Poisson(\gamma_{i,t})\\ & S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\ & N_{i,t+1} = S_{i,t+1} + R_{i,t+1} \end{align*}$$` **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` -- **Integrated likelihood** `$$\mathcal{L}(p, \lambda, \gamma, \phi \mid y_{it}) = \prod_{i=1}^{R} \left[ \sum_{N_{i1} = 0}^{\infty} \cdots \sum_{N_{iT} = 0}^{\infty} \left\{ \left( \prod_{t=1}^{T} Bin(y_{it}; N_{it}, p) \right)\\ \times Pois(N_{i1}; \lambda) \cdot \prod_{t=2}^{T} P_{N_{it}, N_{it+1}} \right\} \right]$$` --- # Dynamic N-mixture model **State process** `$$\begin{align*} & N_{i,1} \sim Poisson(\lambda)\\ & R_{i,t+1} \sim Poisson(\gamma_{i,t})\\ & S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\ & N_{i,t+1} = S_{i,t+1} + R_{i,t+1} \end{align*}$$` **Observation process** `$$\begin{align*} & y_{i,j,t} \sim Binomial(N_{i,t}, p) \end{align*}$$` **Integrated likelihood** `$$\mathcal{L}(p, \lambda, \gamma, \phi \mid y_{it}) = \prod_{i=1}^{R} \left[ \sum_{N_{i1} = 0}^{K} \cdots \sum_{N_{iT} = 0}^{K} \left\{ \left( \prod_{t=1}^{T} Bin(y_{it}; N_{it}, p) \right)\\ \times Pois(N_{i1}; \lambda) \cdot \prod_{t=2}^{T} P_{N_{it}, N_{it+1}} \right\} \right]$$` --- # Transition matrix * Key part of Hidden Markov Models * Involves the transition between abundance at time `\(t\)` and `\(t+1\)` and the demographic parameters * Each element of the transition matrix `\(P\)` represents the transition probability from state `\(N_{it} = j\)` to state `\(N_{it+1} = k\)` with the discrete convolution: `$$P_{jk} = \sum_{c=0}^{\min(j, k)} Bin(c; j, \phi) \cdot Pois(k - c; \gamma)$$` -- `$$\begin{array}{c@{\qquad}c} N_{i,t} & \begin{array}{c} \scriptstyle N_{i,t+1} \\[-0.3ex] \begin{bmatrix} P_{K0} & P_{K1} & P_{K2} & \cdots & P_{KK} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ P_{20} & P_{21} & P_{22} & \cdots & P_{2K} \\ P_{10} & P_{11} & P_{12} & \cdots & P_{1K} \\ P_{00} & P_{01} & P_{02} & \cdots & P_{0K} \end{bmatrix} \end{array} \end{array}$$` --- # Transition matrix .center[ <img src="data:image/png;base64,#images/transmat_extreme.png" width="80%" /> ] --- # Transition matrix <br><br> .center[ <img src="data:image/png;base64,#images/transmat_middle.png" width="4567" /> ] --- # Transition matrix - Optimization * One transition matrix **per time step and per time-series** `\(\Rightarrow\)` Bottleneck `$$P_{jk} = \sum_{c=0}^{\min(j, k)} Bin(c; j, \phi) \cdot Pois(k - c; \gamma)$$` `$$\begin{array}{c@{\qquad}c} N_{i,t} & \begin{array}{c} \scriptstyle N_{i,t+1} \\[-0.3ex] \begin{bmatrix} P_{K0} & P_{K1} & P_{K2} & \cdots & P_{KK} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ P_{20} & P_{21} & P_{22} & \cdots & P_{2K} \\ P_{10} & P_{11} & P_{12} & \cdots & P_{1K} \\ P_{00} & P_{01} & P_{02} & \cdots & P_{0K} \end{bmatrix} \end{array} \end{array}$$` * **Fast, heavy memory implementation:** vectorisation of `\(P\)` over every time steps and batch size * **Time gain:** e.g. from 88 minutes in Bayesian framework (JAGS) to ca. 15 minutes with this implementation (8GB VRAM, 300 epochs) --- # Data simulation * Simulating quadratic relationship between a covariate `\(x\)` and the parameters `\(p\)`, `\(\lambda\)`, `\(\gamma\)`, and `\(\phi\)` * 300 sites, 15 years long time series, 5 repeated counts .center[ <img src="data:image/png;base64,#images/simulated_data.png" width="70%" /> ] --- # Results of the simulation * Simple MLP: 1 hidden layer, 64 hidden units, 300 epochs <!-- * Run time: XXX minutes (CUDA, 8GB VRAM, batch size = 8) --> .center[ <img src="data:image/png;base64,#images/epochs_params_1.png" width="90%" /> ] --- # Next step * So far the architechture of the network is a Multilayer Perceptron * Next step is to use a Convolutional Neural Network with this Loss Function * Potential to use satellite images as input * This will tell us which features of the landscape are enhancing survival and recruitment * Could give real insights on landscape management and provide guidelines for conservation strategies * Use this framework on large spatio-temporal data --- # Limitations * In real life, parametric assumptions are rarely met: the field sampling won't often sample from Poisson and Binomial distribution * Especially true for Recruitment (`\(\gamma\)`) and Survival (`\(\phi\)`), where the input data doesn't contain clear information (unlike abundance `\(N\)`) * Tends to overestimate survival rate over recruitment * Marginalization over `\(K\)`: if too small, biased likelihood, if too big, slow computation/memory heavy --- # Summary * **Neural Dynamic N-mixture model** * Merges hierarchical ecological modeling with deep learning flexibility * Enables fast, parallelizable inference on GPUs * **Interpretability retained** * Demographic parameters (survival, recruitment, detection, abundance) are explicitly inferred from count data * **Performance advantage** * Scales better than classical Bayesian MCMC approaches * Recovers demographic parameters reliably * **Future potential** * Extends demographic rate estimation from count data to applications with computer vision --- # &emsp;&emsp;&emsp;&emsp;Acknowledgements .center[ <img src="data:image/png;base64,#images/acknowledgment.png" width="92%" /> ]